Efficient estimation in the bivariate censoring model and repairing NPMLE

Abstract

The NPMLE in the bivariate censoring model is not consistent for continuous data. The problem is caused by the singly censored observations. In this paper we prove that if we observe the censoring times or if the censoring times are discrete, then a NPMLE based on a slightly reduced data set, in particular, we interval censor the singly censored observations, is asymptotically efficient for this reduced data and moreover if we let the width of the interval converge to zero slowly enough, then the NPMLE is also asymptotically efficient for the original data. We are able to determine a lower bound for the rate at which the bandwidth should converge to zero. Simulation results show that the estimator for small bandwidths has a very goodperformance. The efficiency proof uses a general identity which holds for NPMLE of a linear parameter in convex models. If we neither observe the censoring times nor the censoring times are discrete, then we conjecture that our estimator based on simulated censoring times is also asymptotically efficient.